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Jacobsthal sequence
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(Definition)
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The Jacobsthal sequence is an additive sequence similar to the Fibonacci sequence, defined by the recurrence relation
 , with initial terms  and  . A number in the sequence is called a Jacobsthal number. The first few are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, etc., listed in A001045 of Sloane's OEIS.
The  th Jacobsthal number is the numerator of the alternating sum
(the denominators are powers of two). This suggests a closed form: by putting the series solution over a common denominator and summing the geometric series in the numerator, we obtain two equations, one for even-indexed
terms of the sequence,
and the other one for the odd-indexed terms,
These equations can be further generalized to
The Jacobsthal numbers are named after the German mathematician Ernst Jacobsthal.
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"Jacobsthal sequence" is owned by PrimeFan.
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(view preamble)
| Also defines: |
Jacobsthal number |
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Cross-references: equations, geometric series, summing, solution, series, closed form, powers of two, denominators, alternating sum, numerator, OEIS, number, terms, recurrence relation, Fibonacci sequence, similar, sequence, additive
There is 1 reference to this entry.
This is version 3 of Jacobsthal sequence, born on 2008-06-24, modified 2008-06-29.
Object id is 10720, canonical name is JacobsthalSequence.
Accessed 315 times total.
Classification:
| AMS MSC: | 11B39 (Number theory :: Sequences and sets :: Fibonacci and Lucas numbers and polynomials and generalizations) |
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Pending Errata and Addenda
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